In triangle $ABC,$ $\cot A \cot C = \frac{1}{2}$ and $\cot B \cot C = \frac{1}{18}.$  Find $\tan C.$
Explanation: From the addition formula for tangent,
\[\tan (A + B + C) = \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1 - (\tan A \tan B + \tan A \tan C + \tan B \tan C)}.\]Since $A + B + C = 180^\circ,$ this is 0.  Hence,
\[\tan A + \tan B + \tan C = \tan A \tan B \tan C.\]From $\cot A \cot C = \frac{1}{2},$ $\tan A \tan C = 2.$  Also, from $\cot B \cot C = \frac{1}{18},$ $\tan B \tan C = 18.$

Let $x = \tan C.$  Then $\tan A = \frac{2}{x}$ and $\tan B = \frac{18}{x},$ so
\[\frac{2}{x} + \frac{18}{x} + x = \frac{2}{x} \cdot \frac{18}{x} \cdot x.\]This simplifies to $20 + x^2 = 36.$  Then $x^2 = 16,$ so $x = \pm 4.$

If $x = -4,$ then $\tan A,$ $\tan B,$ $\tan C$ would all be negative.  This is impossible, because a triangle must have at least one acute angle, so $x = \boxed{4}.$